Polynomial maps and polynomial sequences in groups

TL;DR

This paper develops a theory of polynomial maps from semigroups to groups, especially locally nilpotent groups, and applies it to solve Waring's Problem for Heisenberg groups.

Contribution

It introduces a formal theory of polynomial maps in groups and demonstrates its application to a classical problem in group theory.

Findings

Established formal properties of polynomial maps in groups

Solved Waring's Problem for Heisenberg groups

Extended polynomial map theory to locally nilpotent groups

Abstract

This paper develops a theory of polynomial maps from commutative semigroups to arbitrary groups and proves that it has desirable formal properties when the target group is locally nilpotent. We apply this theory to solve Waring's Problem for Heisenberg groups in a sequel to this paper.